Transport in a gravity dual with a varying gravitational coupling constant

We study asymptotically AdS Brans-Dicke (BD) backgrounds, where the Ricci tensor R is coupled to a scalar in the radial dimension, as effective models of metals with a varying coupling constant. We show that, for translationally invariant backgrounds, the regular part of the dc conductivity ${\sigma }_Q$ deviates from the universal result of Einstein-Maxwell-dilaton (EMD) models. However, the shear viscosity to entropy ratio saturates the Kovtun-Son-Starinets (KSS) bound. Similar results apply to more general f(R) gravity models. In four bulk dimensions we study momentum relaxation induced by gravitational and electromagnetic axion-dependent couplings. For sufficiently strong momentum dissipation induced by the former, a recently proposed bound on the dc conductivity $\sigma$ is violated for any finite electromagnetic axion coupling. Interestingly, in more than four bulk dimensions, the dc conductivity for strong momentum relaxation decreases with temperature in the low temperature limit. In line with other gravity backgrounds with momentum relaxation, the shear viscosity to entropy ratio is always lower than the KSS bound. The numerical computation of the optical conductivity reveals a linear growth with the frequency in the limit of low temperature, low frequency and large momentum relaxation. We have also shown that the module and argument of the optical conductivity for intermediate frequencies are not consistent with cuprates’ experimental results, even assuming several channel of momentum relaxation.

Figure 1

Temperature dependence of the dc conductivity Eq. (23) for fixed axion parameter $\alpha =1$ and charge density $\rho =1$. The charge screening parameter $\kappa$ and the BD-coupling parameter $\lambda$ are indicated in the plots. The effect of increasing $\lambda$ for fixed $\kappa$ is similar to increasing $\kappa$ for fixed $\lambda$. The case $\lambda <0$ suggests that the effect of a weaker gravitational coupling $Z<1$ on the dc conductivity is to weaken momentum dissipation. However, we stress that this limit violates the null energy condition Eq. (29) and should be excluded.

Figure 2

Temperature dependence of the dc conductivity Eq. (23) for fixed axion parameter $\alpha =1$ and charge density $\rho =1$. The charge screening parameter $\kappa$ and the BD-coupling parameter $\lambda$ are indicated in the plot. The effect of the BD parameter is similar to the charge screening parameter. Increasing $\lambda$ yields a smaller $r_0^2{c}_0$; as a consequence the first term in Eq. (23), $Y_0=\exp [-\kappa {\alpha }^2/(r_0^2{c}_0)]$, decreases. The bound of [20] is violated for large $\lambda$ even for weak charge screening.

Figure 3

Metric function $c$, Eq. (14), in $d+1=5$ (left column) and $d+1=6$ (right column) bulk dimensions for the model given in Eq. (26). The temperature is indicated in the plots and the charge density is $\rho =1$. The charge screening parameter $\kappa =1$ and the axion parameter $\alpha =1$. The BD-coupling parameter $\lambda$ is indicated in the legend, which refers to all figures. While the boundary conditions for $\lambda <0$ are satisfied, this case leads to violation of the null energy condition.

Figure 4

Zero-frequency conductivity Eq. (23) in $d+1=5$ bulk dimensions in a model with a minimally coupled Maxwell field, Eq. (31). When the BD coupling $\lambda$ is larger than some positive value the dc conductivity at zero temperature is below 1. This effect is due to the background rather than to an axion-dependent Maxwell coupling as in Fig. 1. The charge density is fixed $\rho =1$.

Figure 5

Zero-frequency conductivity Eq. (23) in $d+1=5$ bulk dimensions in a model with a minimally coupled Maxwell field and $V(\mathrm{Tr}X)=0$, Eq. (25). Similar behavior as in Fig. 4 is observed despite the absence of the usual kinetic term in the action ($V=0$). Again, the effective change in the gravitational interaction, through the BD coupling $ZR$, parametrized by ${\lambda }_{\mathrm{eff}}={\alpha }^2\lambda$, allows us to decrease the dc conductivity despite the absence of charge screening, $Y=1$. The charge density is fixed, $\rho =1$.

Figure 6

Zero-temperature [extremal background of Eqs. (13) and (26)] optical conductivity for strong breaking of translational symmetry. The charge screening parameter is $\kappa =1$ and the axion parameter is $\alpha =1$ (blue lines), $\alpha =1.5$ (red lines) and $\alpha =2$ (black continuous lines). The BD-coupling parameter $\lambda$, given in the legend, is fixed close to the maximum value allowed by the background boundary conditions. The extremal charge density is $\rho =1$. The dashed black lines correspond to the linear scaling $\beta \omega$, where $\beta$ is fixed from the lowest frequency point of the numerical data.

Figure 7

Optical conductivity at finite temperature in the background given in Eqs. (13) and (26). Each line corresponds to a fixed temperature, indicated in the legend of the right-hand plots. The dashed-dotted black lines correspond to the quadratic scaling and are fixed in the same way as in Fig. 6. As temperature decreases (blue lines) a slight disagreement is observed. We expect that for near extremal solutions the frequency scaling is given by Eq. (32). The charge density $\rho =1$, $\alpha =1$, $\lambda =0.15$ and $\kappa$ is indicated in the left-hand plots.

Figure 8

Absolute value and argument of the optical conductivity for different values of the parameters. The temperature and charge density are $T=0.05$ and $\rho =1$. The BD model with the couplings of Eq. (26) does not describe the experimental behavior for intermediate frequencies observed for cuprates. The dashed lines are the exponent $\xi$, $|\sigma |={\omega }^a$, $a=1.35-2=-0.65$, and $\arg (\sigma )=\frac{\pi }{2}(2-1.35)=1.01$ Rad, according to experiments.

Figure 9

Shear viscosity to entropy density ratio ($\eta /s$) for the couplings of Eq. (26) in the model defined in Eqs. (13) and (26). The axion parameter $\alpha =1$ and the charge density $\rho =1$.

Figure 10

Shear viscosity to entropy density ratio ($\eta /s$) for the couplings of Eq. (26) in the model defined in Eqs. (13) and (26) but taking $V=0$ and $\dot{V}=0$. The axion parameter $\alpha =1$ and the charge density $\rho =1$. As discussed in Sec. 3b1, for $\kappa =0\Rightarrow Y=1$ (dots) the theory has a single parameter that controls momentum dissipation, namely ${\lambda }_{\mathrm{eff}}=\lambda {\alpha }^2$.

Figure 11

Metric functions $g$ (blackening factor) and $c$, Eq. (14), for two different temperatures: (top row) $T={10}^{-4}$ and (bottom row) $T=0.08$. We fix the charge density $\rho =1$, $\kappa =1$ and ${\lambda }_{\mathrm{eff}}=\lambda {\alpha }^2=0.15$. Each line corresponds to ${\alpha }^2={\lambda }_{\mathrm{eff}}/\lambda$ for the corresponding $\lambda$, which is given in the legends. The legends also refer to the right-hand figures. Moreover, for a fixed temperature and ${\lambda }_{\mathrm{eff}}$, the dc conductivity is different for the two choices of $\lambda$ and $\alpha$. We conclude that, in the presence of $V$ in the action, $\lambda$ and $\alpha$ are two independent parameters associated with the translational symmetry breaking.